The automorphism groups of certain tetravalent metacirculant graphs

Recently, in connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by Phi(1), Phi(2) and Phi(3), have been defined [11]. It has also been shown [11] that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a graph in one of the families Phi(1), Phi(2) or Phi(3). A natural question raised from the result is whether all graphs in these families are non-Cayley, In this paper we determine the automorphism groups of all graphs in the family Phi(2). As a corollary, we show that every graph in Phi(2) is a connected non-Cayley tetravalent metacirculant graph.